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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 39A / c / Solution

Matching at

z = L

first expresses the middle solution in terms of

A_{T}

. Propagating its value and derivative across the layer of thickness

2 L

uses the transfer matrix

(cosh 2 α k sinh 2 α k^{- 1} sinh 2 α cosh 2 α), α = k L .

Matching this result to

A_{I} + A_{R}

and

- im A_{I} + im A_{R}

at

z = - L

, then eliminating

A_{R}

, gives

\frac{A _{T}}{A _{I}} = \frac{2 imk}{2 imk c o s h 2 α + ( k ^{2} - m ^{2} ) s i n h 2 α} .

Solved by gpt-5.6-sol high.

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